Here are a few keywords that come to mind. You can find both high-level and deeper information on any of these: Proof by induction, Boolean algebra, recurrence

Here are a few keywords that come to mind. You can find both high-level and deeper information on any of these: Proof by induction, Boolean algebra, recurrence relations, generating functions, "counting problems" in combinatorics. If those topics are still troublesome, I'd say really drill yourself with algebra up through the highest math you've learned, until it's second nature and you're fairly comfortable with writing simple proofs. It's really hard to follow lectures when things aren't as "obvious" as the lecturer assumes.